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1.7 – Linear Inequalities and Compound Inequalities

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1 1.7 – Linear Inequalities and Compound Inequalities
Properties of Inequalities Addition and Subtraction Property of Inequality If a < b, then a + c < b + c or If a > b, then a + c > b + c If a < b, then a - c < b - c or If a > b, then a - c > b - c Multiplication and Division Property of Inequality c is positive: If a < b, then a β€’ c < b β€’ c or If a > b, then a β€’ c > b β€’ c If a < b, then a/c < b/c or If a > b, then a/c > b/c c is negative: If a < b, then a β€’ c > b β€’ c or If a > b, then a β€’ c < b β€’ c If a < b, then a/c > b/c or If a > b, then a/c < b/c

2 1.7 – Linear Inequalities and Compound Inequalities
Solving Inequalities Examples: 4π‘₯βˆ’9+3π‘₯≀2π‘₯βˆ’5+7π‘₯ βˆ’7 π‘₯+9 β‰₯40+3π‘₯ 7π‘₯βˆ’9≀9π‘₯βˆ’5 βˆ’7π‘₯βˆ’63β‰₯40+3π‘₯ βˆ’2π‘₯≀4 βˆ’10π‘₯β‰₯103 π‘₯β‰₯βˆ’2 π‘₯β‰€βˆ’10.3 -3 -2 -1 9 10 11 βˆ’2, ∞ βˆ’βˆž, βˆ’10.3

3 1.7 – Linear Inequalities and Compound Inequalities
Solving Inequalities Examples: βˆ’2 2βˆ’2π‘₯ βˆ’4 π‘₯+5 β‰€βˆ’24 3 1βˆ’2π‘₯ >8βˆ’6π‘₯ βˆ’4+4π‘₯βˆ’4π‘₯βˆ’20β‰€βˆ’24 3βˆ’6π‘₯>8βˆ’6π‘₯ βˆ’24β‰€βˆ’24 3>8 (1) Lost the variable (1) Lost the variable (2) True statement (2) False statement ∴ Solution: All Reals ∴ Solution: the null set -1 1 -1 1 βˆ’βˆž, ∞ βˆ… or { }

4 Properties of Inequalities Union and Intersection of Sets
1.7 – Linear Inequalities and Compound Inequalities Properties of Inequalities Union and Intersection of Sets The Union of sets A and B 𝐴βˆͺ𝐡 represents the elements that are in either set. The Intersection of sets A and B 𝐴∩𝐡 represents the elements that are common to both sets. Examples: Determine the solution for each set operation. 𝐴: π‘₯|π‘₯β‰₯5 𝐡: π‘₯|3≀π‘₯<12 𝐢: π‘₯|π‘₯<βˆ’1 ) ) 5 3 12 -1 𝐴∩𝐡 𝐴βˆͺ𝐡 𝐡∩𝐢 𝐴βˆͺ𝐢 5, 12 3, ∞ βˆ… βˆ’βˆž, βˆ’1 βˆͺ 5, ∞

5 Compound Inequalities
1.7 – Linear Inequalities and Compound Inequalities Compound Inequalities Example: 7π‘£βˆ’5β‰₯ π‘œπ‘Ÿ βˆ’3π‘£βˆ’2β‰₯βˆ’2 7π‘£βˆ’5β‰₯65 βˆ’3π‘£βˆ’2β‰₯βˆ’2 7𝑣β‰₯70 βˆ’3𝑣β‰₯0 𝑣β‰₯10 𝑣≀0 βˆ’βˆž, 0 βˆͺ 10, ∞

6 Compound Inequalities
1.7 – Linear Inequalities and Compound Inequalities Compound Inequalities Example: 8π‘₯+8β‰₯βˆ’ π‘Žπ‘›π‘‘ βˆ’7βˆ’8π‘₯β‰₯βˆ’79 8π‘₯+8β‰₯βˆ’64 βˆ’7βˆ’8π‘₯β‰₯βˆ’79 8π‘₯β‰₯βˆ’72 βˆ’8π‘₯β‰₯βˆ’72 π‘₯β‰₯βˆ’9 π‘₯≀9 βˆ’9, 9

7 Absolute Value Equations Properties of Absolute Values Equations
1.8 – Absolute Value Equations and Inequalities Absolute Value Equations Properties of Absolute Values Equations 𝒖 =𝒂 π‘Ž<0 No Solution π‘Ž=0 One solution: 𝑒=0 π‘Ž>0 Two Solutions: 𝑒=π‘Ž π‘œπ‘Ÿ 𝑒=βˆ’π‘Ž 𝑒 = π‘š 𝑒=𝑀 or u=βˆ’w 𝑒 = 𝑀 ±𝑒=±𝑀 +𝑒=+𝑀 +𝑒=βˆ’π‘€ βˆ’π‘’=+𝑀 βˆ’π‘’=βˆ’π‘€ 𝑒=𝑀 𝑒=βˆ’π‘€ 𝑒=βˆ’π‘€ 𝑒=𝑀 𝑒=𝑀 𝑒=βˆ’π‘€

8 Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities Absolute Value Equations Examples: βˆ’2𝑛+6 =6 βˆ’2𝑛+6=6 βˆ’2𝑛+6=βˆ’6 βˆ’2𝑛=0 βˆ’2𝑛=βˆ’12 𝑛=0 𝑛=6 𝑛=0, 6 π‘₯+8 βˆ’5=2 π‘₯+8 =7 π‘₯+8=βˆ’7 π‘₯+8=7 π‘₯=βˆ’15 π‘₯=βˆ’1 π‘₯=βˆ’15, βˆ’1

9 Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities Absolute Value Equations Example: 3 3βˆ’5π‘Ÿ βˆ’3=18 3 3βˆ’5π‘Ÿ =21 3βˆ’5π‘Ÿ =7 3βˆ’5π‘Ÿ=βˆ’7 3βˆ’5π‘Ÿ=7 βˆ’5π‘Ÿ=βˆ’10 βˆ’5π‘Ÿ=4 π‘Ÿ=2 π‘Ÿ=βˆ’ 4 5 π‘Ÿ=βˆ’ 4 5 , 2

10 Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities Absolute Value Equations Example: 5 9βˆ’5𝑛 βˆ’7=38 5 9βˆ’5𝑛 =45 9βˆ’5𝑛 =9 9βˆ’5𝑛=βˆ’9 9βˆ’5𝑛=9 βˆ’5𝑛=βˆ’18 βˆ’5𝑛=0 𝑛= 18 5 𝑛=0 𝑛=2, 18 5

11 Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities Absolute Value Equations Example: 2π‘₯βˆ’1 = 4π‘₯+9 2π‘₯βˆ’1=4π‘₯+9 2π‘₯βˆ’1=βˆ’ 4π‘₯+9 βˆ’2π‘₯=10 2π‘₯βˆ’1=βˆ’4π‘₯βˆ’9 π‘₯=βˆ’5 6π‘₯=βˆ’8 π‘₯=βˆ’ 8 6 =βˆ’ 4 3 π‘₯=βˆ’5, βˆ’ 4 3

12 Absolute Value Inequalities Properties of Absolute Values Inequalities
1.8 – Absolute Value Equations and Inequalities Absolute Value Inequalities Properties of Absolute Values Inequalities 𝑒 <π‘Ž βˆ’π‘Ž<𝑒<π‘Ž 𝑒 β‰€π‘Ž βˆ’π‘Žβ‰€π‘’β‰€π‘Ž 𝑒 >π‘Ž 𝑒<βˆ’π‘Ž π‘œπ‘Ÿ 𝑒>π‘Ž 𝑒 β‰₯π‘Ž π‘’β‰€βˆ’π‘Ž π‘œπ‘Ÿ 𝑒β‰₯π‘Ž 10π‘¦βˆ’4 <34 10π‘¦βˆ’4>βˆ’34 10π‘¦βˆ’4<34 10𝑦>βˆ’30 10𝑦<38 𝑦>βˆ’3 𝑦< π‘œπ‘Ÿ 𝑦<3.8 βˆ’3<𝑦<3.8 βˆ’3, 3.8

13 Absolute Value Inequalities Properties of Absolute Values Inequalities
1.8 – Absolute Value Equations and Inequalities Absolute Value Inequalities Properties of Absolute Values Inequalities 𝑒 <π‘Ž βˆ’π‘Ž<𝑒<π‘Ž 𝑒 β‰€π‘Ž βˆ’π‘Žβ‰€π‘’β‰€π‘Ž 𝑒 >π‘Ž 𝑒<βˆ’π‘Ž π‘œπ‘Ÿ 𝑒>π‘Ž 𝑒 β‰₯π‘Ž π‘’β‰€βˆ’π‘Ž π‘œπ‘Ÿ 𝑒β‰₯π‘Ž βˆ’8π‘₯βˆ’3 >11 βˆ’8π‘₯βˆ’3<βˆ’11 βˆ’8π‘₯βˆ’3>11 βˆ’8π‘₯<βˆ’8 βˆ’8π‘₯>14 π‘₯<βˆ’ π‘œπ‘Ÿ π‘₯>1 π‘₯>1 π‘₯<βˆ’ 14 8 βˆ’βˆž, βˆ’ 7 4 βˆͺ 1, ∞ π‘₯<βˆ’ 7 4

14 Absolute Value Inequalities Properties of Absolute Values Inequalities
1.8 – Absolute Value Equations and Inequalities Absolute Value Inequalities Properties of Absolute Values Inequalities 𝑒 <π‘Ž βˆ’π‘Ž<𝑒<π‘Ž 𝑒 β‰€π‘Ž βˆ’π‘Žβ‰€π‘’β‰€π‘Ž 𝑒 >π‘Ž 𝑒<βˆ’π‘Ž π‘œπ‘Ÿ 𝑒>π‘Ž 𝑒 β‰₯π‘Ž π‘’β‰€βˆ’π‘Ž π‘œπ‘Ÿ 𝑒β‰₯π‘Ž 4 6βˆ’2π‘Ž +8≀24 4 6βˆ’2π‘Ž ≀16 6βˆ’2π‘Ž ≀4 6βˆ’2π‘Žβ‰₯βˆ’4 6βˆ’2π‘Žβ‰€4 βˆ’2π‘Žβ‰₯βˆ’10 βˆ’2π‘Žβ‰€βˆ’2 π‘Žβ‰€5 π‘Žβ‰₯1 1β‰€π‘Žβ‰€5 [1, 5]

15 Absolute Value Inequalities Properties of Absolute Values Inequalities
1.8 – Absolute Value Equations and Inequalities Absolute Value Inequalities Properties of Absolute Values Inequalities 𝑒 <π‘Ž βˆ’π‘Ž<𝑒<π‘Ž 𝑒 β‰€π‘Ž βˆ’π‘Žβ‰€π‘’β‰€π‘Ž 𝑒 >π‘Ž 𝑒<βˆ’π‘Ž π‘œπ‘Ÿ 𝑒>π‘Ž 𝑒 β‰₯π‘Ž π‘’β‰€βˆ’π‘Ž π‘œπ‘Ÿ 𝑒β‰₯π‘Ž 9 π‘Ÿβˆ’2 βˆ’10<βˆ’73 9 π‘Ÿβˆ’2 <βˆ’63 π‘Ÿβˆ’2 <βˆ’7 π‘Žπ‘π‘ π‘œπ‘™π‘’π‘‘π‘’ π‘£π‘Žπ‘™π‘’π‘’ π‘π‘Žπ‘›π‘›π‘œπ‘‘ 𝑏𝑒 π‘›π‘’π‘”π‘Žπ‘‘π‘–π‘£π‘’ βˆ΄π‘›π‘œ π‘ π‘œπ‘™π‘’π‘‘π‘–π‘œπ‘›

16 Absolute Value Inequalities Properties of Absolute Values Inequalities
1.8 – Absolute Value Equations and Inequalities Absolute Value Inequalities Properties of Absolute Values Inequalities 𝑒 <π‘Ž βˆ’π‘Ž<𝑒<π‘Ž 𝑒 β‰€π‘Ž βˆ’π‘Žβ‰€π‘’β‰€π‘Ž 𝑒 >π‘Ž 𝑒<βˆ’π‘Ž π‘œπ‘Ÿ 𝑒>π‘Ž 𝑒 β‰₯π‘Ž π‘’β‰€βˆ’π‘Ž π‘œπ‘Ÿ 𝑒β‰₯π‘Ž 5 3π‘›βˆ’2 +6β‰₯51 5 3π‘›βˆ’2 β‰₯45 3π‘›βˆ’2 β‰₯9 3π‘›βˆ’2β‰€βˆ’9 3π‘›βˆ’2β‰₯9 3π‘›β‰€βˆ’7 3𝑛β‰₯11 π‘›β‰€βˆ’ 7 3 𝑛β‰₯ 11 3 βˆ’βˆž, βˆ’ 7 3 βˆͺ , ∞ π‘›β‰€βˆ’ π‘œπ‘Ÿ 𝑛β‰₯ 11 3


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